A dictionary training algorithm was proposed for spare representation of images and its convergence was proved.The geometrical explanation of the algorithm is to approximate the hyperspherical cap with least hyperplanes.The algorithm clustered the error vectors of each step,and signed the cluster center as new atoms which made the dictionary more suitable for spare representation of samples.Compared with the traditional algorithm,the new one has higher adaptability,lower requirement of sample number and dictionary size,higher convergence rate,and lower complexity.Finally,the experiment of compressive sensing and denoising demonstrates that dictionary training by this algorithm has good effect.